PHYSICAL REVIEW E 88, 062708 (2013)

Forces due to curving protofilaments in microtubules Shirish Vichare,1 Ishutesh Jain,2 Mandar M. Inamdar,1,* and Ranjith Padinhateeri2,† 1

Department of Civil Engineering, Indian Institute of Technology Bombay, Mumbai 400076, India 2 Department of Biosciences and Bioengineering, Indian Institute of Technology Bombay, Mumbai 400076, India (Received 19 August 2013; published 10 December 2013) Microtubules consist of 13 protofilaments arranged in the form of a cylinder. The protofilaments are composed of longitudinally attached tubulin dimers that can exist in either a less curved state [GTP-bound tubulin (T)] or a more curved state [GDP-bound tubulin (D)]. Hydrolysis of T into D leaves the straight and laterally attached protofilaments of the microtubule in a mechanically stressed state, thus leading to their unzipping. The elastic energy in the unzipping protofilaments can be harnessed by a force transducer such as the Dam1-kinetochore ring complex in order to exert pulling force on chromosomes during cell division. In the present paper we develop a simple continuum model to obtain this pulling force as a function of the mechanical properties of protofilaments and the size of the Dam1-kinetochore ring. We also extend this model to investigate the role played by the T subunits found at the plus end of the microtubule (the T cap) on the mechanical stability of microtubules. DOI: 10.1103/PhysRevE.88.062708

PACS number(s): 87.10.Pq, 83.10.Ff

I. INTRODUCTION

Microtubules are major constituents of a cell’s cytoskeleton, i.e., the scaffolding that governs structural stability and the shape of living cells. They are crucially involved in cell division, cell motility, and provide a network for intracellular transport [1]. Microtubules are polymers made of tubulin subunits, which are typically bound to GTP or GDP. Many such subunits connect longitudinally to each other with the same orientation to form a single protofilament. Typically, 13 such protofilaments also bond laterally with each other to form a cylindrical microtubule [2]. Protofilaments made of GDP-bound tubulin (D) are known to have an intrinsically curved conformation, whereas protofilaments composed of GTP-bound tubulin (T) are known to be relatively straight [3]. One end of the microtubule, the plus end, is highly dynamic compared to the other end and grows by the addition of T subunits. The T subunits found at the plus end of the microtubule comprise what is generally known as a T cap [2]. However, the GTP molecules affixed to the tubulin can hydrolyze to GDP and eventually change the structural conformation of the whole protofilament. It is a well known principle in mechanics that constraining the kinematic degree(s) of freedom of a mechanical system leads to the production of the corresponding generalized force(s) [4]. Since the D form of the protofilament tends to be curved, its confinement in a straight configuration in a cylindrical microtubule can generate significant lateral and longitudinal stress [5]. Moreover, if the lateral restraints are not sufficient, the protofilaments can unzip from each other, thus leading to structural instability or catastrophe of the microtubule [6–9]. However, if a sufficient amount of constraint is provided in the form of a structural T cap, then microtubule integrity can still be maintained [7–9]. A microtubule with a T cap at its plus end has an internal power struggle between the T and D forms: The T form tends to remain straight and conforms to the cylindrical geometry of the microtubule,

* †

[email protected] [email protected]

1539-3755/2013/88(6)/062708(11)

whereas the D form strives for exactly the opposite. In such a scenario, an adequate T cap can provide the necessary restraining force on the D filament at the T-D interface, thus preventing potential catastrophe [5,7–9]. There may also be alternative mechanisms by which D protofilaments can be constrained and thus stabilized. For example, the unzipping D protofilaments can be restrained by coupling them to ringlike proteins such as the Dam1-kinetochore complex [10,11]. In this case, the reaction force generated at the protofilament-ring interface can be used to drive chromosome segregation during cell division [12,13]. Keeping in sight this constraint-force complementarity, the main goal of this paper is to understand, using simple models, how curling D protofilaments can generate force when restrained. In particular, we are interested in getting simple analytical insights into force generation at protofilament–Dam1-kinetochore contact and at the T-D interface. It has been experimentally observed that curved unzipping microtubule protofilaments can generate force due their coupling with a ringlike protein complex known as the Dam1– DASH-kinetochore complex via depolymerization [13]. For example, Westermann et al. observed that a Dam1 ring moves processively on depolymerizing microtubule protofilament ends [10]. In another set of experiments, Asbury and coworkers showed that the Dam1 ring can stay attached to the microtubule tip, harness energy from microtubule disassembly, and move under opposing forces of up to 3 pN [11,14]. Further, by studying force production by depolymerizing microtubules with a laser trap, Grishchuk et al. concluded that microtubules generate up to 5 pN force per protofilament via a power-stroke mechanism, where the curling protofilaments exert a direct physical force on the attached optical bead [15]. More recently, Hendricks et al. demonstrated that dynein motors positioned at the microtubule plus end with a laser trap forced a delay in microtubule catastrophe by exerting tension on individual protofilaments [16]. In addition to these experimental studies, there have been a number of theoretical studies investigating, either directly or indirectly, force generation in curling protofilaments [7,8,17–24]. For instance, in the context of the

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Dam1-kinetochore complex, by accounting for the bending and interprotofilament interactions, Molodtsov et al. developed a detailed micromechanical model to quantify the amount of force that can be generated by a curving protofilament [18]. In contrast, Armond and Turner studied the problem of Dam1 ring movement using the Fokker-Planck equation by considering it as a biased diffusion process, without explicitly taking into account the mechanics of microtubule protofilaments [22]. Elsewhere, a detailed model by Liu and Onuchic took into account electrostatic and specific interactions between the protofilaments and the Dam1 ring, in addition to bending and lateral interactions, and computationally obtained the translational dynamics of the Dam1 ring [19]. On a related front, detailed micromechanical models by Molodtsov et al. [8] and Hunyadi and J´anosi [20] demonstrated that, at least energetically, even a single layer of a T cap can provide sufficient resistance to curling D protofilaments and prevent microtubule catastrophe. Similarly, a detailed mechanochemical computational model by VanBuren et al. also highlighted the role of internal conformational forces in governing the overall microtubule kinetics [17]. Using a more simplified description, but by explicitly taking filament dynamics into account with a Langevin dynamics description, Zapperi and Mahadevan developed a simple model to explain how the interplay between time scales for polymerization, hydrolysis, and filament unzippering determines microtubule growth kinetics [23]. More recently, by using the principles of structural mechanics, Jin and Ru developed a simple analytical model to study how the combination of microtubule length, interfilament adhesion, and compressive load can lead to protofilament unzippering [25]. A similar analytical model, but without an explicit description of lateral interactions, was developed earlier by J´anosi et al. [7] With this brief preamble we now arrive at the main point of this paper. As motivated earlier, the primary focus of this paper is to understand, using simple modeling, how the application of constraints leads to force generation in microtubule protofilaments. To this end we will particularly focus on two problems. Specifically, we will first develop a simple analytical model to study the force exerted by a curling protofilament that is attached to its neighboring filaments by short-range nonspecific interactions and constrained at its tip by connection to a Dam1-like ring. Although Molodtsov et al., for example, have already developed a detailed micromechanical model in this regard [8,18], the complexity of the model preempts any analytical insights. In this work we develop a continuum model for this purpose and obtain analytical results—in certain cases even closed-form formulas—that also give numerical estimates in the same range as the more complicated numerical simulations. Additionally, by extending this model to incorporate the T cap, we also investigate the role of the T cap in determining microtubule integrity. Although some earlier works discussed the role of the T cap in microtubule stability, they have a few deficiencies. For example, the computational models for a microtubule with a cap by Molodtsov et al. [8] and Hunyadi and J´anosi [20] are not capable of providing simple analytical insights. In contrast, the analytical model by J´anosi et al. [7], though simple and elegant, suffers from the drawback that it does not explicitly account for the lateral interactions and potential protofilament

unzippering [20]. We hope to fill some of these lacunae by the work in the current paper. The outline of the paper is as follows. In Sec. II we develop a continuum model for force generation by protofilaments due to the restraint provided by the Dam1-kinetochore complex. In Sec. III we extend the model developed in Sec. II to incorporate the T cap and examine its role in maintaining structural integrity of the microtubule. We follow up these two sections with a brief discussion and conclusion in Sec. IV. II. FORCE GENERATION BY CURLING PROTOFILAMENTS WHEN RESTRAINED BY THE DAM1-KINETOCHORE COMPLEX A. Model

We assume a 13-fold rotational symmetry for microtubules and consider each of the 13 protofilaments as identical and subject to the exact same loading or mechanical constraint [8,18]. Due to this rotational symmetry it will suffice to model the mechanical behavior of only one protofilament. We model a fully D protofilament as a long, slender, elastic rod with uniform bending stiffness B and intrinsic curvature c˜0 . Due to the assumed radial symmetry, lateral interactions of the protofilament with its two neighbors are depicted as adhesive energy per unit length α of the filament with a rigid wall (Fig. 1). As can be seen below, when α is less than the bending energy per unit length of a straight protofilament B c˜02 /2, the protofilament catastrophically peels off (unzips) from its neighbors. In this work we assume that the mechanical equilibration of the protofilaments happens at a faster time scale when compared with the depolymerization time scale and therefore we neglect the possible length change of the protofilament due to depolymerization. First, we describe the mechanics of a D protofilament whose unzipping is halted due to the fixed displacement constraint applied at its end by the Dam1-kinetochore complex (see Fig. 1). Without going into the details of the interactions

FIG. 1. (Color online) Mechanical model for a peeling D protofilament that is restricted at its free end. The constraint of the two translational degrees of freedom of the end node should result in resisting forces Fx and Fy that are represented by an effective force F acting at an angle β with respect to the horizontal. The lateral interaction of the protofilament with its neighbors (modeled as a rigid wall) is represented by an effective adhesive energy per unit length α. The protofilament is tethered by a pin support at the other end to prevent global translation. Here Rx and Ry represent the position of the tip of the protofilament with respect to a completely straight unpeeled conformation. A section of the filament at length s depicting internal force F and bending moment M is shown in the inset.

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between the Dam1 ring and the protofilament, the basic principles of mechanics would dictate that since the horizontal and vertical displacements of the filament tip are constrained by the Dam1 ring its mechanical presence can be replaced by two forces Fx and Fy or, identically, a force F˜ acting at an angle β with respect to the horizontal (see Fig. 1). The other end of the protofilament is tethered, as shown in Fig. 1, to prevent rigid body translation of the complete microtubule in the horizontal direction. The geometry of the peeled protofilament of length L˜ is provided by the arclength parameter s˜ and tangent angle θ . The coordinates of the protofilament tip are provided by R˜ y and R˜ x : Both R˜ y and R˜ x are zero when the filament is completely straight and unpeeled. Note that while R˜ y is the difference between the radii of the Dam1 ring R˜ ring and microtubule R˜ MT , R˜ x captures the relative horizontal movement of the protofilament tip from its straight unpeeled position. The elastic-adhesive energy for the above model can be written as   2  L˜  ˜ B L dθ cos θ (s)ds H= − c˜0 ds + Fx L˜ − 2 0 ds 0    L˜ 1 2 ˜ + Fy sin θ (s)ds − (1) B c˜ − α L, 2 0 0

current formulation, all three quantities F˜ , β, and θ0 (tangent angle at the filament tip) are unknown. Using the scales

where Fx and Fy are the horizontal and vertical components of the force acting at the protofilament tip, s˜ is the arclength parameter, and θ (˜s ) is the angle between the tangent vector at s˜ and the horizontal (see Fig. 1). Note that this is a variable end-point problem with L˜ another unknown in addition to θ (s) [26,27]. The resulting Euler-Lagrange equation from this energy is given by

As a consequence of the nondimensionalization, the model for protofilament essentially has only one parameter c0 ; the bending modulus and lateral interaction energy are unity. Equation (8) can be rearranged in the form

Bθ  (˜s ) − F˜ sin(θ + β) = 0,

(2)

where θ  = dθ , Fx = F˜ cos β, and Fy = F˜ sin β. Following d s˜ the standard procedure [28,29] in solving such problems in elastica theory, we rewrite the above equation as B  2 θ (˜s ) + F˜ cos(θ + β) = C1 . 2

(3)

The constant C1 in Eq. (3) is obtained to be equal to α + F˜ cos β by noting the following boundary conditions at s = 0: θ (0) = 0,

B  2 θ (0) = α. 2

(4)

Here the first condition is a reflection of the fact that the de-adhered portion of protofilament is horizontal, whereas the second condition results from extremization of energy H over the peeled length L˜ [26,27]. Further noting that the curvature ˜ = c˜0 (no bending ˜ is θ  (L) at the protofilament tip (at s˜ = L) ˜ moment) and introducing θ0 = θ (L), we obtain the following relation between θ0 , F˜ , and β: F˜ cos β = 12 B c˜02 − α + F˜ cos(θ0 + β).

 L0 ≡

B , F0 ≡ α α

(6)

for the length and force, respectively, and the corresponding dimensionless quantities R˜y L˜ R˜x s˜ , L≡ , Ry ≡ , Rx ≡ , L0 L0 L0 L0 F˜ c0 ≡ c˜0 L0 , F ≡ , F0 s≡

(7)

Eq. (3) can now be written in the following form: 1  θ (s)2 2

+ F cos(θ + β) = 1 + F cos β.

(8)

The boundary conditions at s = 0 and s = L can also be rewritten as 1  θ (0)2 2

θ (0) = 0,

= 1, θ  (L) = c0 ,

θ (L) = θ0 . (9)

√  dθ (s) = ± 2 1 + F [cos β − cos(β + θ )] ds

(10)

to provide a differential relation between θ and s. We now make a simplifying assumption that the function θ (s) is single valued with respect to s. However, it may be noted that it is not too difficult to extend this formalism even if θ is not a monotonic function of s by appropriate identification of inflection points [θ  (s) = 0] and subsequent juxtaposition of identical curves [29]. The simplest possible way in which the protofilament can be constrained by the Dam1 ring is shown in Fig. 1, where the tip is fixed at given values of Rx and Ry ; the corresponding forces Fx = F cos β and Fy = F sin β are unknown. Since we are only interested in obtaining the pulling force exerted by the protofilament on the Dam1 ring we take π/2  β  −π/2. Similarly, we disallow any displacement below the plane of the substrate because it implies that the symmetric protofilaments interpenetrate. Using the positive square root from Eq. (10), we can write equations for Ry and Rx as

(5)

This equation is particularly instructive because the term on the left-hand side is the pulling force in the horizontal direction on the ring. Depending on whether θ0 + β is either greater or less than π/2, we will get a driving force Fx = F˜ cos β that is either less or greater than B c˜02 /2 − α, the energy release per unit length when the protofilament unzips. Note that, as per the 062708-3



L

Ry =

sin θ (s)ds 0

 θ0 sin θ 1 dθ, (11) = √ √ 1 + F [cos β − cos(θ + β)] 2 0  L Rx = [1 − cos θ (s)]ds 0

1 = √ 2



θ0 0

√

1 − cos θ dθ. (12) 1 + F [cos β − cos(θ + β)]

VICHARE, JAIN, INAMDAR, AND PADINHATEERI 4

(a)

π/4 π/6 0 -π/6 -π/4

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4

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(b)

π/4 π/6 0 -π/6 -π/4

3.5 3 2.5 F cosβ

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2

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1

1

0.5

0.5

0

β

0 0

0.5

1

1.5 Ry

2

2.5

3

0

0.5

1

1.5

2

2.5

Rx

FIG. 2. (Color online) Pulling force F cos β exerted by a protofilament on the kinetochore for different values of Ry and Rx . Different combinations of F and β provide a range of values of (a) Ry and (b) Rx as per Eqs. (11) and (12), respectively. Thus, for a Dam1 ring of radius Rring = Ry + RMT , the value for the pulling force and the corresponding horizontal tip displacement Rx can be obtained by consulting these two figures. These estimates can be used to obtain peeled protofilament length L from Fig. 3. Note that all the quantities are dimensionless as per Eq. (8); the value of dimensionless curvature used is c0 = 1.6. For parameters relevant to microtubules, as described in the text, the unit force is ≈ 1.5 pN and the unit length is ≈ 32 nm.

To obtain θ0 we use nondimensionalized form of Eq. (5) resulting in 1 2 c + F cos(θ0 + β) = 1 + F cos β 2 0   co2 /2 − 1 −1 cos β − − β. ⇒ θ0 = ±2nπ ± cos F

(13)

We thus have Eqs. (11) and (12) along with Eq. (13), from which we can, in principle, obtain the two unknowns F and β. In practice, it is much easier and mechanically equivalent (when temperature T = 0) to assume a range of values of F and β and use them to get the corresponding values of Ry and Rx . Moreover, the peeled-off length of the protofilament is given by  θ0 1 1 dθ. (14) L= √ √ 1 + F [cos β − cos(θ + β)] 2 0 B. Results

Considering the simplest case of n = 0 in Eq. (13), we have   co2 2 − 1 −1 θ0 = cos cos β − − β. (15) F For this solution to exist it is clear that c2 2 − 1  −1, 1  cos β − o F which implies that

 cos β c02 2 − 1  F ≡ c02 2 − 1  F cos β = Fx . 1 + cos β 1 + cos β (16) This condition puts a clear lower bound on the magnitude of the force acting at the protofilament tip if the spontaneous unzipping of the protofilament is to be prevented. Note that the pulling force is dictated by the differential between the curling

energy and the lateral interaction energy. For the filament to spontaneously start unzipping, c02 /2 − 1 has to be positive, i.e., the curling or curvature energy is greater than the adhesive energy [20]. This results in a very simple but illuminating criterion on the value of F that can be exchanged between the protofilament and the kinetochore. We now numerically integrate Eqs. (11) and (12) using the computational software MATHEMATICA to obtain Ry , Rx , and L for a set of F and β values in conjunction with the expression for θ0 obtained in Eq. (15). Since the primary quantity of interest is the pulling force on the Dam1 ring F cos β and its dependence on the ring radius Ry = Rring − RMT , we present the results as shown in Fig. 2. We have assumed that −π/2  β  π/2 to ensure that Fx is essentially a pulling force on the Dam1 ring. As can be seen from Fig. 1, β > 0 corresponds to the force pushing down on the protofilament, attempting to straighten it out. Correspondingly in Fig. 2, for β > 0 and for relatively large F , the values of Rx and Ry are low. This argument can be turned around to infer that when the ring radius is small, the pulling force on the ring can be large. When β < 0, the vertical component of the force tries to destabilize the filament, whereas the horizontal component prevents the filament from peeling off. This means that for large negative β, it is not possible to get very small Rx and Ry . For a given Ry , multiple estimates of Fx = F cos β correspond to different values of Rx (horizontal displacement of protofilament tip). In other words, if the ring radius or Ry is given, the pulling force can be obtained from Figs. 2(a) and 2(b), with Fig. 2(b) reflecting the accompanying value of Rx . Now that we have a general understanding of the relation between Rx , Ry , and the pulling force F cos β, we also compute the influence of the these parameters on the length of the protofilament that is peeled off from the substrate; the results are shown in Fig. 3. Consistent with our previous results, when β is positive and the force F is relatively large, less of the protofilament peels off. In other words, when the ring radius is small, as expected, a small number of protofilaments break their lateral bonds. In contrast, when

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β

F cosβ

2.5 2 1.5 1 0.5 0 0

0.5

1

1.5

2

2.5

3

3.5

4

L

FIG. 3. (Color online) Peeled off length of the protofilament for different combinations of F and β values. For typical microtubule parameters, as described in the text, the unit force is ≈1.5 pN and the unit length is ≈32 nm.

β is negative there is a fewfold increase in the length of unzipped protofilament. To summarize, a small ring radius leads to less peeling of protofilaments and is more capable of producing larger pulling forces on the Dam1 ring. When the ring radius is larger, there is a significantly higher peeling of the protofilament and force transduction of the stored potential energy in the curved D protofilament is relatively low. We do not directly observe the nonmonotonicity of the pulling force as a function of the ring radius as seen by Molodtsov et al. [18]. One of the reasons for this, we think, is that, unlike Molodtsov et al., the simple representation of lateral interactions that we have used in our model does not have nonmonotonicity with respect to lateral spacing between protofilaments. Moreover, we have an additional variable Rx , the horizontal position of the protofilament, in our model that Molodtsov et al. do not explicitly account for. To obtain numerical values of the pulling force we first estimate values of the scaling parameters that are given in Eq. (6). The curling energy per dimer is approximately 4kB T ≈ 16.4 pN nm at room temperature, whereas the lateral interaction energy per dimer is around 3kB T ≈ 12.3 pN nm [20]. The tubulin size is approximately d = 8 nm, whereas the spontaneous curvature of the protofilament is around c˜0 = 1/20 nm−1 [18,20]. Using these numbers, the values of B and α are estimated as 1 B c˜02 d 2

estimates of the dimensionless pulling force Fx would be between 1 and 3, which is approximately 1.5–4.5 pN in actual units. Even from this idealized model, the numbers we can obtain are roughly similar to those observed in the experiments and simulations [15,18]. Note that, in principle, for a given Ry , there are infinite allowable values of F cos β according to the theory, each value corresponding to a particular Rx . However, we simply demonstrate that an appropriate choice of Rx [with Rx comparable to Ry (≈0.15), which results in a small peel off length β close to zero and θ0 < π/2] can provide a decent estimate of the pulling force for a given ring radius. Though, according to our model, a protofilament can exert a relatively large force for a ring of small radius Ry and a comparably small value of Rx (β > 0), what we additionally learn is that the protofilament can exert a comparable pulling force on a ring of larger Ry when the value of Rx is comparable by making β < 0. The consequence of having β < 0 is that the end angle θ0 becomes greater than π/2 and the radial force Fy on the ring acts inward, as opposed to outward when β > 0. So if the mechanical interactions at the point of contact between the protofilament and the ring can support this force, then, in principle, even rings of larger radii can be utilized to transduce pulling forces comparable to their smaller radii counterparts. When β > 0, we can use a very simple approximation to understand the behavior of the curves in Figs. 2 and 3. Assuming that the shape of the rod is approximately an arc of a circle of length L with curvature c0 , the bending moment at any arclength s is given as M(s) = −

where Fy = F cos β and Fx = F sin β. Using the criterion developed earlier in Eq. (9), we note that the amount of length L peeled off the substrate will be such that 1  θ (0)2 2

1 [M(0) 2

+ c0 ]2 = 1,

(20)

which implies that √ F [− cos β + cos(c0 L + β)] + c0 = 2 c0  √ 2c0 − c02 + F cos β 1 . − β + cos−1 ⇒L= c0 F (21)

(17)

The length and force scales then become  B ≈ 32 nm, F0 ≡ α ≈ 1.54 pN. L0 ≡ (18) α The outer radius of the microtubule R˜ MT is approximately 12.5 nm, whereas the inner radius R˜ ring of the Dam1 ring is around 17.5 nm [30]. Hence the value of R˜y equals 5 nm, which correspond to dimensionless Ry ≈ 0.15. As can be seen from Fig. 2, there are multiple possible values of Fx = F cos β for this Ry , corresponding to different inputs of Rx . Reasonable

= 1,

thus

= 16.4 pN nm ⇒ B ≈ 1640 pN nm2 ,

αd = 12.3 pN nm ⇒ α ≈ 1.54 pN.

Fy Fx (sin Lc0 − sin sc0 ) − (cos sc0 − cos Lc0 ), c0 c0 (19)

Using this value of L, we can easily obtain the values of Ry and Rx using the following formula: 1 (1 − cos Lc0 ) − uy , c0 1 Rx = L − sin Lc0 − ux . c0

Ry =

(22) (23)

Here uy and ux are the displacement of the tip of the filament of length L from its preferred configuration, with curvature c0 , in the directions of Fx and Fy , respectively, which are obtained

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by the use of Castigliano’s second theorem [31]: ∂U ux = , ∂Fx

PHYSICAL REVIEW E 88, 062708 (2013) 3.5 (a) 3

∂U , (25) ∂Fy where U is the potential energy in the filament, which is obtained using Eq. (19) as  1 L U= M(s)2 ds. (26) 2 0

2.5

uy =

F cosβ

(24)

π/4 π/6

2 1.5 1 0.5

We can clearly see from Fig. 4 that the simple solution obtained from this linearized analysis matches the complete geometrically nonlinear solution when β > 0.

0 0

0.05

0.1

0.15 Rx

0.2

0.25

3.5 (b)

C. Maximal pulling force generated by the protofilament depends on the details of the interaction between the protofilament and Dam1 ring

−1 . cos β + sin φ

0.3

β

(28)

Further, as can be seen from Fig. 5(a), θ0 is simply π/2 − (β − φ). Finally, we are left with just one unknown β, which can be, for a given Rx , numerically obtained from Eqs. (11) and (28). It can be seen that when φ = 0, the pulling force F cos β reduces to c02 /2 − 1, the effective energy release per unit length of the filament. For nonzero φ, the calculated F cos β is the minimum pulling force that the filament has to exert on the ring in order to stay in mechanical equilibrium when the interaction between the filament and the ring is similar to dry friction [dotted lines in Fig. 5(c) for small (φ = 0.1) and moderate

F cosβ

2.5

c02 + F cos(π/2 + φ) = 1 + F cos β. (27) 2 The value of the resultant force F at the point of contact then becomes F =

π/4 π/6

3

2 1.5 1 0.5 0 0

0.1

0.2

0.3

0.4

0.5

Ry 3.5 (c)

π/4 π/6

3

β

2.5 F cosβ

In the results obtained so far, we did not concern ourselves with the nature of the interaction between the Dam1 ring and the protofilament. Instead, we went about our calculations by assuming that, given a displacement constraint at the protofilament end, the ring-filament interaction was capable of providing appropriate reaction forces such the system is in mechanical equilibrium; similar considerations were made when forces were instead applied at the protofilament tip and the corresponding displacements calculated. However, in reality, the exact nature of the protofilament-ring interaction is likely to provide some bounds on the pulling force that a protofilament can exert on the Dam1 ring. To provide some insight into how these interactions can limit the production of the pulling force, we now analyze a special case when the reaction between the protofilament and the ring is taken to be similar to dry friction [32]: Ft  μFn , where Ft = F sin φ, Fn = F cos φ, and μ = tan−1 φ are the tangential force, normal force, and coefficient of friction at the point of contact between the protofilament and Dam1 ring, respectively. An extreme case is when the ring has an impending slippage with respect to the protofilament. If the impending slippage of the ring is to the left [Fig. 5(a)] then Eq. (8) becomes

c02 /2

β

2 1.5 1 0.5 0 0

0.1

0.2

0.3 L

0.4

0.5

0.6

FIG. 4. (Color online) Comparison between estimates for the kinetochore pulling force obtained from elastica theory using Eqs. (11), (12), and (14) and the simplified results obtained from Castigliano’s second theorem (dotted curves). It can be seen that the simplified theory gives a very good match with the exact results. The value of dimensionless curvature used is c0 = 1.6.

(φ = 0.5) friction]. In the other limit when the impending slippage of the ring with respect to the protofilament is to the right [Fig. 5(b)], the entire analysis can be repeated as before by replacing φ with −φ. In this case the calculated pulling force F cos β represents the maximum pulling force that can act between the ring and the protofilament after which the entire assembly will cease to be in mechanical equilibrium [dotted

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(c)

1.4

φs = -0.5

1.2 F cosβ

1 0.8 0.6 φs = -0.1

0.4

φs = 0.1

0.2

φs = 0.5

0 0

0.2 0.4 0.6 0.8

1 Ry

1.2 1.4 1.6 1.8

2

FIG. 5. (Color online) Forces acting at the tip of the protofilament if the interaction between the protofilament and kinetochore is akin to dry friction. The impending slippage of the ring with respect to the protofilament can be either (a) to the left or (b) to the right. The angle of friction, i.e., the angle between the resultant force and the normal to the filament at the point of contact, is taken as φ. It is clear from (a) that θ0 + β − φ = π/2. When the impending slippage is to the right (b), then φ will be replaced with −φ. (c) Corresponding pulling force F cos β exerted by the protofilament on the ring for different values of Ry and φ. The dotted lines correspond to φ = 0.1 and 0.5 when the impending slippage of the ring is to the left, whereas the solid lines are for φ = −0.1 and − 0.5 when the impending slippage of the ring is to the right. The solid and dotted curves provide bounds on the pulling force on the ring and highlight the importance of the ring-protofilament interactions. The value of dimensionless curvature used is c0 = 1.6.

lines in Fig. 5(c) for small (φ = 0.1) and moderate (φ = 0.5) friction]. The dotted and the solid lines in Fig. 5(c) essentially provide lower and upper bounds, respectively, on the pulling force that can be exchanged between the protofilament and the ring. This is very similar, in spirit, to the requirement of maximum and minimum upward force to maintain sliding equilibrium of a block on a wedge when the friction between the block and the wedge is specified [32]. It is amply clear from various evidence that the interaction between protofilaments and the Dam1 ring is very different from the simplistic frictional interaction that we have modeled here [19,33]. However, this simple calculation highlights a very important fact that the specifics of the interaction between the ring and the protofilament establish bounds on the otherwise unrestricted force transmission between them [compare Figs. 2(b) and 5(c)]. This observation highlights the fact that the actual nature of the interaction between the protofilament and the Dam1 ring can be greatly influential in deciding the force that is generated.

III. THE GTP CAP MODEL

So far, we have discussed how the restraint provided by a force transducer, such as the Dam1-kinetochore complex, could harness the potential energy of protofilaments having only GDP subunits and generate force. As we mentioned in the Introduction, GTP cap can also act as a constraint and resist the unzipping of GDP protofilaments, thus maintaining the structural integrity of microtubules. To understand this quantitatively, we extend the model developed earlier by incorporating the physics of GTP cap.

FIG. 6. (Color online) Simple one-filament model for a microtubule with a T cap. The D part (length LD ) of the protofilament attempts to peel off the lateral interaction α and in turn is resisted by the T cap (length LT ), which prefers to stay straight. At the interface between T and D, vertical force FI and bending moment MI will be generated to maintain displacement and slope continuity between the two parts.

Consider an infinitely long protofilament that has two parts: One part is made of D subunits while the other part is made of T subunits (see Fig. 6). The D part will try to curve and peel off while pulling the T part along with it. Depending on the dimensionless curvature c0 , the overall system is likely to reach a mechanical equilibrium in the configuration shown in Fig. 6. Using the same nondimensionalization as earlier [Eq. (7)] by assuming the same lateral interaction energy and bending modulus for both T and D (Fig. 6), the equations of equilibrium [Eq. (8) with β = ±π/2] can be written for the D and T filaments, respectively, as follows:

and

1  θ (s )2 2 D D

− FI sin θD (sD ) = kD

(29a)

1  θ (s )2 2 T T

+ FI sin θT (sT ) = kT .

(29b)

Here sT and sD are the arclength parameters for the T and D filaments, respectively, with their origins as shown in Fig. 6, and FI is the internal force at the interface of T and D parts. One can easily find that the constants have values kD = 1 and kT = 1 by noting that at mechanical equilibrium [Eq. (9)], 1  θ (0)2 2 D

= 12 θT (0)2 = 1,

θD (0) = θT (0) = 0.

(30)

Here we have implicitly assumed that the lateral interaction energy for both T (αT ) and D (αD ) protofilaments are equal (α). However, this can be easily modified by using a new parameter γ = αT /αD > 1, the resulting expressions for which are shown in the Appendix. Additionally, the tangent continuity at the point of contact between D and T dictates that θD (LD ) = −θT (LT ) = θ0 . Also, the moment continuity at the interface between D and T implies that θD (LD ) − c0 = θT (LT ) = −MI . Using Eq. (29), we obtain the following relations for MI and θ0 : 1 (−MI 2

+ c0 )2 − FI sin θ0 = 1,

1 (−MI )2 2

− FI sin θ0 = 1

(31) (32)

for D and T, respectively. These two equations can be easily solved to give c02 8 − 1 c0 . (33) MI = , sin θ0 = 2 FI

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VICHARE, JAIN, INAMDAR, AND PADINHATEERI 0.825

6

FI LT

5.5

0.82

5 4.5

0.815

4

LT

FI

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3.5

0.81

3 2.5

0.805

2 1.5

0.8 4.5 4.55 4.6 4.65 4.7 4.75 4.8 4.85 4.9 c0

FIG. 7. Internal force FI at the T-D interface and the corresponding √ peeled off length for the T protofilament LT when 4.55  c0 < 2 6. Note that LT does not vary significantly.

Equation (29) can be rewritten as θD (sD )2 = 2[1 + FI sin θ (sD )], θT (sT )2 = 2[1 − FI sin θ (sT )].

(34)

The only unknown parameter left in the problem is the internal force FI at the D-T interface, which can be obtained by observing that the vertical height δi has to be the same for both D and T (Fig. 6). The corresponding equation is  θs  θ0 2 sin θ sin θ 1 dθ − √ dθ √ √ √ 1 − F sin θ 1 + FI sin θ 2 0 2 0 I  θs  θ0 sin θ sin θ 1 dθ ⇒ dθ = √ √ √ 1 + FI sin θ 1 − FI sin θ 2 0 0  θ0 sin θ 1 dθ for θs = sin−1 , = √ FI 1 + FI sin θ 0 2 −1 c0 8 − 1 . (35) θ0 = sin FI Here the nonmonotonicity of T (see Fig. 6) is understood by noting that according to Eq. (34), θT (sT ) = θs indicates a point of inflection for the T filament, i.e., θT (sT ) = √ 0. This equation will admit solutions only when 4.55  c0  2 6; the force FI as a function of c0 is shown in Fig. 7. We interpret this to mean that when the value of c0 is less than 4.55, the microtubule will be internally stable even in the presence of the √ smallest of the caps. In contrast, when c0 is greater than 2 6 ≈ 4.9, the protofilament will be internally unstable, irrespective of the size of the cap. For any intermediate values of c0 , it can be seen that the critical cap size can be obtained from the expression  θs 1 2 dθ LT = √ √ 1 − FI sin θ 2 0  θ0 1 1 dθ, (36a) +√ √ 1 + FI sin θ 2 0  θ0 1 1 dθ. (36b) LD = √ √ 1 + FI sin θ 2 0

Paradoxically, the critical length LT for the cap size required to keep the microtubule stable marginally decreases with increasing value of c0 (Fig. 7). This can be understood by noting that the smaller the value of LT , the greater the stiffness is and hence the higher its resistance is for the peeling D filament. However, the value of the force FI , at the interface, to keep the system in equilibrium increases with c0 , as expected (Fig. 7). The upper limit for c0 can be obtained using the approximation of small θ due to which sin θ ≈ θ in Eq. (35). The resulting equation then becomes √    √ 64 + 2 −24 + c02 c0 2 2 = . (37) √ 3FI2 48 2FI2 √ The solution for this equation gives c0 = 2 6, thus providing an upper limit for the value of c0 beyond which Eq. (35) has no real solution. To get some more analytical insight into this problem in terms of closed-form solutions, we can also perform a simple calculation for the small-θ limit to obtain the deformed shape and energy as a function of LT and LD . Using simple principles of structural mechanics [31], the displacement δT and rotation θT at the point of contact between D and T as observed for the T filament is given by L3T L2 (38a) F I − T MI , 3 2 L2 θT = T FI − LT MI . (38b) 2 Similarly, when θ is small, the linearized governing equation of equilibrium for the D filament will be given by δT =

θD (s) − FI = 0, θD (0) = 0,

θD (LD ) − c0 = −MI . (39)

This linear equation can be readily solved to provide the following equation for θD (s): θD (s) = c0 s − FI LD s − MI s +

FI s 2 . 2

(40)

Using this, the values of displacement δD and rotation θD at the point of intersection between T and D, as observed for the D filament, are given as c0 L2D L2 L3 − D F I − D MI , 2 3 2 L2D F s − LD M I . θD = c0 LD − 2 δD =

(41a) (41b)

Now, because of the compatibility of the displacement (δT = δD ) and rotation (θT = −θD ) at the point of T-D intersection, the values of FI and MI can be evaluated from Eqs. (38) and (41) as

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c0 LD LT FI = 6 , (LD + LT )3

2  c0 LD LD − LD LT + 4L2T MI = . (LD + LT )3

(42) (43)

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The total elastic and adhesive energy of the system can then easily be obtained as

 c02 L2D L2D + 2LD LT + 4L2T Utotal (LT ,LD ) = 2(LD + LT )3 1 − c02 LD + (LT + LD ). (44) 2 Using a parameter r = LT /LD , the total energy can be rewritten as  2  c (1 + 2r + 4r 2 ) 1 2 c − + 1 + r . U˜ total (LD ,r) = LD 0 2(1 + r)3 2 0 (45) Upon minimizing the energy with respect to LT (or r), we obtain the following equation in r: −c02 (1 − 2r)2 + 2(1 + r)4 = 0. 2(1 + r)4

(46)

A relevant solution to this equation  √ −3 2c0 + c02 c0 (47) r = −1 + √ + √ 2 2 provides r, which essentially determines the ratio of LT /LD at mechanical equilibrium. However, it may be noted that this does not uniquely provide the individual values of either LT or LD because there is no solution for the two simultaneous equations that is obtained by minimizing Utotal with respect to both LD and LT . Physically this means that the energy minima with respect to LD and LT , though very close to each other, do not exactly match. Nevertheless, the values of r obtained from this procedure compare extremely well with √ the values obtained using the exact Eq. (36) when c0 → 2 6 ≈ 4.899, the limit when the small θ approximation is quite accurate. For example, when c0 = 4.88, using Eqs. (35) and (36), we can obtain the values of LT ≈ 0.707 and LD ≈ 0.190, which gives r = LT /LD ≈ 3.728, very close to the estimate obtained from Eq. (47) (r ≈ 3.698). Similarly, using these values of LD and LT in Eqs. (42) and (43) provides estimates of FI ≈ 5.44 and MI ≈ 2.44, which are very close to the values obtained from Eq. (35) (FI ≈ 5.51) and Eq. (33) (MI = c0 /2 = 2.44). This simple mechanical analysis thus provides some quick insights and an internal consistency check for our more detailed solution. IV. DISCUSSION AND CONCLUSION

We have developed a simple single protofilament elasticaadhesive model [34] to study the unpeeling of microtubule protofilaments and the resulting force generation. Using the model we have analyzed the force produced by a curling protofilament, its consequences for kinetochore pulling during cell division, and T cap requirement for maintaining structural stability. We have obtained a compact description of the relations between the force and displacement of protofilament tips in terms of a single dimensionless parameter c0 , the protofilament spontaneous curvature. Moreover, as can be seen from Eq. (16), we obtained a closed-form lower bound for the net value of the contact force between the protofilament and the Dam1 ring. Further, our calculations using Eqs. (11), (12),

and (14), shown in Figs. 2 and 3, also made it possible to obtain the pulling force for the kinetochore in terms of the constraint parameters Ry and Rx . These calculations are supported by a simplified analytical estimate, obtained from Castigliano’s second theorem, which shows a very good match with the fully nonlinear elastica theory (see Fig. 4). The force per protofilament, as per our numerical estimates, is of the order of 5 pN per protofilament, which is comparable to the forces observed in earlier experiments and computations [15,18]. We have also analyzed a special case by assuming a frictionlike interaction between the Dam1 ring and the protofilament to obtain near-analytical results for the pulling force, while emphasizing the importance of ring-protofilament interactions (Fig. 5). Upon analyzing the role of the T cap by quantitative√ modeling, we have derived a simple criterion 4.55  c0  2 6 to decide the stability of the microtubule. When c0 is less than 4.55 the protofilament will not spontaneously peel off even with a small amount of the √ protofilament T cap. In contrast, when c0 is greater than 2 6 the filament will spontaneously peel off irrespective of the size of the T cap. We also performed a simple analytical √ calculation to obtain the energy landscape when c0 → 2 6 and the deflections of the protofilament are very small. We would like to note here that according to the numerical estimates in this work, the value of c0 is approximately 1.6, which is less than 4.55. This, even after providing for some margin of error, implies that microtubules can be internally stabilized even with a single T cap; this finding is consistent with detailed numerical simulation of Hunyadi and J´anosi [20]. However, in the absence of the T cap the protofilaments will spontaneously peel off, making the microtubule unstable. The clear drawbacks of this model are as follows. We model only one protofilament by assuming symmetry. However, the inherent stochasticity in the filament dynamics can lead to instabilities and possible symmetry breaking [35]. Moreover, the current symmetry assumption also preempts inclusion of the seam, typically seen in microtubules. As a result, any possible effect of the seam will be absent in our results [2]. We assume hard lateral interactions between a protofilament and its neighbors, i.e., when bound to its neighbors, the protofilament is exactly straight and catastrophically peels from the surface according to energetic requirements. Given that the length-scale defining range of lateral interactions is very small (a few nanometers [18,20]), it is quite reasonable to model the interactions in terms of adhesive energy per unit length α. Nevertheless, such modeling does not explicitly allow for any lateral bending or stretching [5,36]. More importantly, this model is strictly a T = 0 model, i.e., the thermal fluctuations are neglected. Further, even after assuming full 13-fold symmetry, strictly speaking, it is not a priori rigorously clear that the protofilaments will stay planar because the problem is geometrically nonlinear and an out-of-plane bending solution may still be possible. However, we are not aware of any experimental estimate of out-of-plane bending and twist coefficients. Nevertheless, a simple back-of-the-envelope calculation suggests that if these three coefficients are equal then the twisting and out-of-plane bending effects cancel out and we are left with only in-plane bending [37]. In the other limit, when the twisting modulus is very large compared to the bending moduli, out-of-plane bending, which leads to torsional energy, will be energetically

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unfavorable. Moreover, if the mechanical interactions at the protofilament-ring interface are such that they restrict twisting rotation, it would be difficult for the protofilaments to bend out of plane and twist. Despite these drawbacks, the most important contribution that is made in this work lies in obtaining simple and compact formulas, and scaling relations. It is quite clear that these results would be less accurate than the corresponding detailed micromechanical models [8,18,20]. Nevertheless, as seen earlier, the current model provides estimates that are both qualitatively and quantitatively similar to the ones obtained from these significantly more complicated models. In our opinion, this fact, combined with the compactness of the results, is the real strength of the current work. ACKNOWLEDGMENTS

I.J. acknowledges support through a fellowship from CSIR, India. R.P. acknowledges support from IYBA, DBT India. M.M.I. acknowledges funding from a seed grant from IIT Bombay. APPENDIX

If the lateral energy for the T protofilament and D protofilament is taken as αT and αD , respectively, then the √  2

θs 0

1 sin θ −√ √ γ − FI sin θ 2



θ0 0

main equations of equilibrium for the two parts of the filaments can be rewritten as θD (sD )2 = 2[1 + FI sin θ (sD )],

(A1)

θT (sT )2 = 2[γ − FI sin θ (sT )],

(A2)

where γ = αT /αD . The equations of equilibrium at the interface then become 1 (−MI + c0 )2 2 1 (−MI )2 2

− FI sin θ0 = 1,

(A3)

− FI sin θ0 = γ

(A4)

for D and T, respectively. These two equations can be solved to give  2 4 2 2 −1 4 − 4c0 + c0 − 8γ − 4c0 γ + 4γ θ0 = sin , 8c02 FI MI =

−2 + c02 + 2γ . 2c0

When γ = 1, these expressions reduce to the ones derived in Sec. III. As done earlier, the final unknown FI is obtained simply by equating the vertical height from the T and D sides at the T-D interface. This equation is

1 sin θ =√ √ γ + FI sin θ 2

 0

θ0

sin θ , √ 1 + FI sin θ

(A5)

where θs = sin−1 (γ /FI ) and θ0 is as defined earlier. As done in the main text, we will obtain an expression for the critical value of c0 beyond which the microtubule will be unstable irrespective of the cap size. To that we linearize the condition at small θ , where we can approximate sin θ with θ . The resulting equation can be written as √

  64c03 γ 3/2 − 2 −2 + 2γ + c02 c04 + 4(−1 + γ )2 − 4c02 (1 + 5γ ) √ 48 2FI2 c03 √

  64c03 + 2 2 − 2γ + c02 c04 + 4(1 − γ )2 − 4c02 (5 + γ ) = . (A6) √ 48 2FI2 c03

This equation can be solved to provide the critical value of c0critical beyond which the microtubule will be unconditionally unstable as  √ √ √ c0critical = 2(−1 + γ + 2 1 + γ + γ ). (A7)

When γ = 1, i.e., the lateral interaction energies for T√and D are identical, the expression for c0critical simplifies to 2 6, the same value obtained in Sec. III. When T lateral interactions critical are increases beyond √ larger than those of T (γ > 1), c0 2 6, as expected.

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